We discuss recent work, including joint with with Emmanuel Kowalski and Philippe Michel, and possibly others, where new bounds on complete exponential sums were proved, which were then applied to analytic number theory problems. We will also discuss some general heuristics for how useful complete exponential sum estimates will be for a particular problem

I will explain the analogy between trace functions over finite fields defined by exponential sums and certain classical functions on the complex numbers defined by integrals of exponentials. There are close analogies, largely due to Katz, that sometimes allow one to guess results in one domain from results in the other. For instance, many important properties of Kloosterman sums are related to facts about Bessel functions. I will explain some of these correspondences, and how to use them to understand exponential sums